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A NOTE ON OSTROWSKI LIKE INEQUALITIES

B.G. PACHPATTE

57 Shri Niketan Colony

Near Abhinay Talkies

Aurangabad 431 001

(Maharashtra) India

EMail: bgpachpatte@hotmail.com

c 2000 Victoria University

ISSN (electronic): 1443-5756

241-04 volume 6, issue 4, article 114,

2005.

Received 13 December, 2004; accepted 23 August, 2005.

Communicated by: A. Sofo

Abstract

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Abstract

The aim of this note is to establish new Ostrowski like inequalities by using a fairly elementary analysis.

2000 Mathematics Subject Classification: 26D10, 26D15.

inequality.

Contents

1 Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 Main Results

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

References

A Note on Ostrowski Like

Inequalities

B.G. Pachpatte

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1.

Introduction

In an elegant note [ 5 ], A.M. Ostrowski proved the following interesting and useful inequality (see also [ 3 , p. 468]):

(1.1) f ( x ) −

1 b − a

Z b a f ( t ) dt ≤

"

1

4

+ x − a + b

2

( b − a )

2

2

#

( b − a ) k f

0 k

, for all x ∈ [ a, b ] , where f : [ a, b ] ⊆ entiable on ( a, b ) , whose derivative f

R

0 k f

0 k

= sup x ∈ ( a,b )

| f

0

( x ) | < ∞ .

R is continuous on [ a, b ] and differ-

: ( a, b ) →

R is bounded on ( a, b ) , i.e.,

In the last few years, the study of such inequalities has been the focus of great attention to many researchers and a number of papers have appeared which

deal with various generalizations, extensions and variants, see [ 2 , 3 , 6 ] and the

references given therein. Inspired and motivated by the recent work going on

related to the inequality ( 1.1

), in the present note , we establish new inequalities of the type ( 1.1

) involving two functions and their derivatives. An interesting

feature of our results is that they are presented in an elementary way and provide new estimates on these types of inequalities.

A Note on Ostrowski Like

Inequalities

B.G. Pachpatte

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2.

Main Results

Our main result is given in the following theorem.

Theorem 2.1. Let entiable on ( a, b ) f, g : [ a, b ] →

R be continuous functions on [ a, b ] and differ-

, whose derivatives f

0

, g

0

: ( a, b ) →

R are bounded on ( a, b ) , i.e., k f

0 k

= sup x ∈ ( a,b )

| f

0

( x ) | < ∞ , k g

0 k

= sup x ∈ ( a,b )

| g

0

( x ) | < ∞ .

Then

(2.1) f ( x ) g ( x ) −

1

2

2 ( b

1

− a

{| g ( x ) | k f

0 k

) g ( x )

Z a b f ( y ) dy + f ( x )

+ | f ( x ) | k g

0 k

}

"

1

4

+

Z b g ( y ) dy a x − a + b

2

( b − a )

2

2 #

( b − a ) , for all x ∈ [ a, b ] .

Proof. For any x, y ∈ [ a, b ] we have the following identities:

(2.2) f ( x ) − f ( y ) =

Z x f

0

( t ) dt y and

(2.3) g ( x ) − g ( y ) =

Z x g

0

( t ) dt.

y

Multiplying both sides of ( 2.2

) and ( 2.3

) by

g ( x ) and f ( x ) respectively and adding we get

(2.4) 2 f ( x ) g ( x ) − [ g ( x ) f ( y ) + f ( x ) g ( y )]

A Note on Ostrowski Like

Inequalities

B.G. Pachpatte

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= g ( x )

Z x y f

0

( t ) dt + f ( x )

Z x y g

0

( t ) dt.

Integrating both sides of ( 2.4

) with respect to

y over [ a, b ] and rewriting we have

(2.5) f ( x ) g ( x ) −

=

2 ( b

1

1

2 ( b − a )

− a )

Z b

Z b

Z a g ( x ) b f ( y ) dy + f ( x ) g ( y ) dy a g ( x )

Z x y a f

0

( t ) dt + f ( x )

Z x y g

0

( t ) dt dy.

From ( 2.5

) and using the properties of modulus we have

1 f ( x ) g ( x ) −

2 ( b − a )

=

1

2 ( b − a )

1

2 ( b − a ) g ( x )

Z

{| g ( x ) | k f

0 k

∞ a b f ( y ) dy + f ( x )

Z a b g ( y ) dy

Z b

{| g ( x ) | k f

0 k

| x − y | + | f ( x ) | k g

0 k

| x − y |} dy a

+ | f ( x ) | k g

0 k

}

"

( x − a )

2

+ ( b − x )

2

#

2

=

1

2

{| g ( x ) | k f

0 k

+ | f ( x ) | k g

0 k

}

"

1

4

+ x − a + b

2

( b − a )

2

2 #

( b − a ) .

The proof is complete.

Remark 1. We note that, by taking g ( x ) = 1 and hence g

0

( x ) = 0 in Theorem

2.1

, we recapture the well known Ostrowski’s inequality in ( 1.1

).

A Note on Ostrowski Like

Inequalities

B.G. Pachpatte

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Integrating both sides of ( 2.5

) with respect to

x over [ a, b ] , rewriting the resulting identity and using the properties of modulus, we obtain the following

Grüss type inequality:

(2.6)

1 b − a

Z b f ( x ) g ( x ) dx − a

1 Z b Z b

2 ( b − a )

2 a a b

1

− a

Z a b f ( x ) dx b

1

− a

Z a b g ( x ) dx

{| g ( x ) | k f

0 k

+ | f ( x ) | k g

0 k

} | x − y | dy dx.

For other inequalities of the type ( 2.6

), see the book [ 3 ], where many other

references are given.

A slight variant of Theorem

2.1

is embodied in the following theorem.

Theorem 2.2. Let f, g, f

0

, g

0 be as in Theorem

2.1

. Then

1 Z b

(2.7) f ( x ) g ( x ) −

+ f ( x ) a g ( x ) f ( y ) dy b −

Z b a a

1 b − a

1 Z b g ( y ) dy + f ( y ) g ( y ) dy b − a k f

0 k

∞ a k g

0 k

"

( x − a )

3

+ ( b − x )

3

#

,

3 for all x ∈ [ a, b ] .

Proof. From the hypotheses, the identities ( 2.2

) and ( 2.3

) hold. Multiplying the

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Inequalities

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left and right sides of ( 2.2

) and ( 2.3

) we get

(2.8) f ( x ) g ( x ) − [ g ( x ) f ( y ) + f ( x ) g ( y )] + f ( y ) g ( y )

=

Z x f

0

( t ) dt

Z x g

0

( t ) dt .

y y

Integrating both sides of ( 2.8

) with respect to

y over [ a, b ] and rewriting we have

1 Z b

(2.9) f ( x ) g ( x ) −

+ f ( x ) a g ( x ) f ( y ) dy b − a

Z b a

=

1 b − a

1 Z b g ( y ) dy + a y f ( y ) g ( y ) dy b − a

Z b Z x a f

0

( t ) dt

Z x y g

0

( t ) dt dy.

From ( 2.9

) and using the properties of modulus we obtain

=

1 Z b f ( x ) g ( x ) − g ( x ) f ( y ) dy b − a a

Z b

1 Z b

1

1 b − a

+ f ( x ) g ( y ) dy + f ( y ) g ( y ) dy b − a a b − a a k f

0 k

∞ k g

0 k

∞ k f

0 k

∞ k g

0 k

Z b

| x − y a

"

( x − a )

3

|

2 dy

+ ( b − x )

3

#

.

3

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Inequalities

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The proof is complete.

Remark 2. Integrating both sides of ( 2.9

) with respect to

x over [ a, b ] , rewriting the resulting identity, using the properties of modulus and by elementary calculations we get

(2.10)

1 b − a

Z b f ( x ) g ( x ) dx − a

1 b − a

Z b a

1 Z b f ( x ) dx g ( x ) dx b − a a

1

12

( b − a )

2 k f

0 k

∞ k g

0 k

.

Here, it is to be noted that the inequality ( 2.10

) is the well known ˇ

inequality (see [ 4 , p. 297]).

A Note on Ostrowski Like

Inequalities

B.G. Pachpatte

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References

[1] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. Pure

and Appl. Math., 31 (2000), 379–415.

[2] S.S.DRAGOMIR

AND

Th.M. RASSIAS (Eds.), Ostrowski Type Inequali-

ties and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht , 2002.

CARI ´

AND

A.M. FINK, Inequalities for

Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

CARI ´

AND

A.M. FINK, Classical and New

Inequalities in Analysis, Kluwer Academic Publishers, Drodrecht, 1993.

[5] A.M. OSTROWSKI, Über die Absolutabweichung einer differentiebaren

Funktion von ihrem Integralmitelwert, Comment. Math. Helv., 10 (1938),

226–227.

[6] B.G. PACHPATTE, On a new generalization of Ostrowski’s inequality, J.

Inequal. Pure and Appl. Math., 5(2) (2004), Art. 36. [ONLINE: http:

//jipam.vu.edu.au/article.php?sid=378 ]

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